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Theorem tpid3g 3722
Description: Closed theorem form of tpid3 3723. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tpid3g (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 2766 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 3mix3 1170 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
32a1i 9 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)))
4 abid 2177 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4imbitrrdi 162 . . . . 5 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}))
6 dftp2 3656 . . . . . 6 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2256 . . . . 5 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7imbitrrdi 162 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9 eleq1 2252 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
108, 9mpbidi 151 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1110exlimdv 1830 . 2 (𝐴𝐵 → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
121, 11mpd 13 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 979   = wceq 1364  wex 1503  wcel 2160  {cab 2175  {ctp 3609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-tp 3615
This theorem is referenced by:  rngmulrg  12652  srngmulrd  12663  lmodscad  12681  ipsmulrd  12693  ipsipd  12696  topgrptsetd  12713
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